Guillaume Mercère

Ziad Alkhoury, Mihály Petreczky, Guillaume Mercère

It is known that in general, \emph{frozen equivalent} (Linear Parameter-Varying) LPV models, \emph{i.e.}, LPV models which have the same input-output behavior for each constant scheduling signal, might exhibit different input-output behavior for non-constant scheduling signals. In this paper, we provide an analytic error bound on the difference between the input-output behaviors of two LPV models which are frozen equivalent. This error bound turns out to be a function of both the speed of the change of the scheduling signal and the discrepancy between the coherent bases of the two LPV models. In particular, the difference between the outputs of the two models can be made arbitrarily small by choosing a scheduling signal which changes slowly enough. An illustrative example is presented to show that the choice of the scheduling signal can reduce the difference between the input-output behaviors of frozen-equivalent LPV models.

Guillaume Mercère

In many practical cases, the engineer has access to prior knowledge like rough values of the DC-gain or the main time constant of the system. In order to improve the accuracy of subspace-based identification techniques using the model Markov parameters, we derive in this short paper the direct links between these impulse response coefficients and this prior information. The next step will consist in introducing this prior knowledge explicitly in Kung's algorithm thank to dedicated equality and equality constraints.

José A. Ramos, Guillaume Mercère

Recently \cite{Ramos2017a} presented a subspace system identification algorithm for 2-D purely stochastic state space models in the general Roesser form. However, since the exact problem requires an oblique projection of $Y_f^h$ projected onto $W_p^h$ along $\widehat{X}_f^{vh}$, where $W_p^h= \begin{bmatrix}\widehat{X}_p^{vh} \\ Y_p^h \end{bmatrix}$, this presents a problem since $\{\widehat{X}_p^{vh},\widehat{X}_f^{vh}\}$ are unknown. In the above mentioned paper, the authors found that by doing an orthogonal projection $Y_f^h/Y_p^h$, one can identify the future horizontal state matrix $\widehat{X}_f^{h}$ with a small bias due to the initial conditions that depend on $\{\widehat{X}_p^{vh},\widehat{X}_f^{vh}\}$. Nevertheless, the results on modeling 2-D images were very good despite lack of knowledge of $\{\widehat{X}_p^{vh},\widehat{X}_f^{vh}\}$. In this note we delve into the bias term and prove that it is insignificant, provided $i$ is chosen large enough and the vertical and horizontal states are uncorrelated. That is, the cross covariance of the state estimates $x_{r,s}^{h}$ and $x_{r,s}^{v}$ is zero, or $P_{hv}=0_{n_x\times n_x}$ and $P_{vh}=0_{n_x\times n_x}$. Our simulations use $i=30$. We also present a second iteration to improve the state estimates by including the vertical states computed from a vertical data processing step, i.e., by doing an orthogonal projection $Y_f^v/Y_p^v$. In this revised algorithm we include a step to compute the initial states. This new portion, in addition to the algorithm presented in \cite{Ramos2017a}, forms a complete 2-D stochastic subspace system identification algorithm.

Ziad Alkhoury, Mihály Petreczky, Guillaume Mercère

In this paper, we introduce and study important properties of the transformation of Affine Linear Parameter-Varying (ALPV) state-space representations into Linear Fractional Representations (LFR). More precisely, we show that $(i)$ state minimal ALPV representations yield minimal LFRs, and vice versa, $(ii)$ the input-output behavior of the ALPV represention determines uniquely the input-output behavior of the resulting LFR, $(iii)$ structurally identifiable ALPVs yield structurally identifiable LFRs, and vice versa. We then characterize LFR models which correspond to equivalent ALPV models based on their input-output maps. As illustrated all along the paper, these results have important consequences for identification and control of systems described by LFRs.